Question

Jim is in a gambling casino that charges $4 per chance to roll

an honest die. He will be paid, in dollars, the number of dots
on the face of the die. Find his mathematical expectation.
Round to the nearest penny.

Answer 1

To find Jim's mathematical expectation, we calculate the expected value by multiplying the value of each outcome by its probability and summing them up. In this case, the mathematical expectation is $3.50 (rounded to the nearest penny).

Step-by-step explanation:

Mathematical Expectation in Gambling Casino

To find Jim's mathematical expectation, we need to calculate the expected value. The expected value is obtained by multiplying the value of each outcome by its corresponding probability and then summing them up. In this case, the possible outcomes are the numbers on the face of the die (1, 2, 3, 4, 5, 6), and each outcome has a probability of 1/6 since the die is honest. The value of each outcome is the same as the number on the face of the die. Therefore, the mathematical expectation can be calculated as follows:

E(X) = (1/6 x 1) + (1/6 x 2) + (1/6 x 3) + (1/6 x 4) + (1/6 x 5) + (1/6 x 6)

E(X) = 3.5

Therefore, Jim's mathematical expectation is $3.50 (rounded to the nearest penny).

Answer 2

$3.50

Step-by-step explanation:

There are six equally likely outcomes when rolling a fair die, with each outcome having a probability of 1/6. Let X be the random variable representing the amount Jim wins, then:

X = 1, 2, 3, 4, 5, or 6, each with probability 1/6.

The mathematical expectation (or expected value) of X, denoted E(X), is the sum of each outcome multiplied by its probability:

E(X) = (1)(1/6) + (2)(1/6) + (3)(1/6) + (4)(1/6) + (5)(1/6) + (6)(1/6)

= 21/6

= 3.5

Therefore, Jim's mathematical expectation is $3.50 per roll, which means on average he can expect to win $3.50 per roll in the long run.